The perimeter of the triangle ABC is 78 cm. Find the lengths of the sides of this triangle

The perimeter of the triangle ABC is 78 cm. Find the lengths of the sides of this triangle if AB: BC = 3: 4, and BC: AC = 2: 3.

Since we have two relations where the BC appears, we will express it:
BC = 4AB / 3, BC = 2AC / 3, equate the obtained expressions and get that AC = 2AB, it is convenient to rewrite everything through AB, but first we denote AB as x, then:
AB = x, AC = 2x, BC = 4x / 3, and our perimeter is 78 cm, then:
x + 2x + 4x / 3 = 78,
13x / 3 = 78,
13x = 234,
x = 18.
It follows from this that AB = 18 cm, AC = 36 cm, BC = 24 cm.
Answer: AB = 18 cm, AC = 36 cm, BC = 24 cm.